1. Discrete Random Variables
6.1
Discrete Random Variables

2. Random Variables
A random variable is a numeric measure of the outcome of a probability experiment
Random variables reflect measurements that can change as the experiment is repeated
Random variables are denoted with capital letters, typically using X (and Y and Z …)
Values are usually written with lower case letters, typically using x (and y and z ...)

3. Examples (Random Variables)
Tossing four coins and counting the number of heads
The number could be 0, 1, 2, 3, or 4
The number could change when we toss another four coins
Measuring the heights of students
The heights could change from student to student

4. Discrete Random Variable
A discrete random variable is a random variable that has either a finite or a countable number of values
A finite number of values such as {0, 1, 2, 3, and 4}
A countable number of values such as {1, 2, 3, …}
Discrete random variables are designed to model discrete variables (see section 1.2)
Discrete random variables are often “counts of …”

5. Example (Discrete Random Variable)
The number of heads in tossing 3 coins (a finite number of possible values)
There are four possible values – 0 heads, 1 head, 2 heads, and 3 heads
A finite number of possible values – a discrete random variable
This fits our general concept that discrete random variables are often “counts of …”

6. Discrete Random Variables
Other examples of discrete random variables
The possible rolls when rolling a pair of dice
A finite number of possible pairs, ranging from (1,1) to (6,6)
The number of pages in statistics textbooks
A countable number of possible values
The number of visitors to the White House in a day

7. Continuous Random Variable
A continuous random variable is a random variable that has an infinite, and more than countable, number of values
The values are any number in an interval
Continuous random variables are designed to model continuous variables (see section 1.1)
Continuous random variables are often “measurements of …”

8. Example (Continuous Random Variable)
An example of a continuous random variable
The possible temperature in Chicago at noon tomorrow, measured in degrees Fahrenheit
The possible values (assuming that we can measure temperature to great accuracy) are in an interval
The interval may be something like (–20,110)
This fits our general concept that continuous random variables are often “measurements of …”

9. Continuous Random Variables
Other examples of continuous random variables
The height of a college student
A value in an interval between 3 and 8 feet
The length of a country and western song
A value in an interval between 1 and 15 minutes
The number of bytes of storage used on a 80 GB (80 billion bytes) hard drive
Although this is discrete, it is more reasonable to model it as a continuous random variable between 0 and 80 GB

10. Probability Distribution
The probability distribution of a discrete random variable X relates the values of X with their corresponding probabilities
A distribution could be
In the form of a table
In the form of a graph
In the form of a mathematical formula

11. Probability Distribution
If X is a discrete random variable and x is a possible value for X, then we write P(x) as the probability that X is equal to x
Examples
In tossing one coin, if X is the number of heads, then P(0) = 0.5 and P(1) = 0.5
In rolling one die, if X is the number rolled, then P(1) = 1/6

12. Probability Distribution
Properties of P(x)
Since P(x) form a probability distribution, they must satisfy the rules of probability
0 ≤ P(x) ≤ 1
Σ P(x) = 1
In the second rule, the Σ sign means to add up the P(x)’s for all the possible x’s

13. Probability Distribution
An example of a discrete probability distribution
All of the P(x) values are positive and they add up to 1

14. NOT a Probability Distribution
An example that is not a probability distribution
Two things are wrong
P(5) is negative
The P(x)’s do not add up to 1

15. Probability Histogram
A probability histogram is a histogram where
The horizontal axis corresponds to the possible values of X (i.e. the x’s)
The vertical axis corresponds to the probabilities for those values (i.e. the P(x)’s)
A probability histogram is very similar to a relative frequency histogram

16. Probability Histogram
An example of a probability histogram
The histogram is drawn so that the height of the bar is the probability of that value

17. Mean of a Probability Distribution
The mean of a probability distribution can be thought of in this way:
There are various possible values of a discrete random variable
The values that have the higher probabilities are the ones that occur more often
The values that occur more often should have a larger role in calculating the mean
The mean is the weighted average of the values, weighted by the probabilities

18. Mean of a Discrete Random Variable
The mean of a discrete random variable is
μX = Σ [ x • P(x) ]
In this formula
x are the possible values of X
P(x) is the probability that x occurs
Σ means to add up these terms for all the possible values x

19. Mean [ x • P(x) ] Example of a calculation for the mean
Add: = 2.5
The mean of this discrete random variable is 2.5

20. Law of Large Numbers
The mean can also be thought of this way (as in the Law of Large Numbers)
If we repeat the experiment many times
If we record the result each time
If we calculate the mean of the results (this is just a mean of a group of numbers)
Then this mean of the results gets closer and closer to the mean of the random variable

21. Expected Value
The expected value of a random variable is another term for its mean
The term “expected value” illustrates the long term nature of the experiments – as we perform more and more experiments, the mean of the results of those experiments gets closer to the “expected value” of the random variable

22. Variance
The variance of a discrete random variable is computed similarly as for the mean
The mean is the weighted sum of the values
μX = Σ [ x • P(x) ]
The variance is the weighted sum of the squared differences from the mean
σX2 = Σ [ (x – μX)2 • P(x) ]
The standard deviation, as we’ve seen before, is the square root of the variance … σX = √ σX2

23. Variance The variance formula σX2 = Σ [ (x – μX)2 • P(x) ]
can involve calculations with many decimals or fractions
An equivalent formula is
σX2 = [ Σ x2 • P(x) ] – μX2
This formula is often easier to compute

24. Good News!
The variance can be calculated by hand, but the calculation is very tedious
Whenever possible, use technology (calculators, software programs, etc.) to calculate variances and standard deviations
See Handout

25. Summary
Discrete random variables are measures of outcomes that have discrete values
Discrete random variables are specified by their probability distributions
The mean of a discrete random variable can be interpreted as the long term average of repeated independent experiments
The variance of a discrete random variable measures its dispersion from its mean

26. Determine whether the random variable is discrete or continuous
Determine whether the random variable is discrete or continuous. State the possible values of the random variable.
The amount of rain in Seattle during April.
The number of fish caught during a fishing tournament
The number of customers arriving at a bank between noon and 1pm
The time required to download a file from the internet

27. Determine whether the distribution is a discrete probability distribution.X
P(x)
100 .1
200
.25
300 .2
400 .3
500

28. In the following probability distribution, the random variable X represents the number of activities a parent of a K-5th grade student is involved in X
P(x)
.035 1
.074 2
.197 3
.320 4
.374
Verify that this is a discrete probability distribution
b) Draw a probability histogram

29. X
P(x)
.035 1
.074 2
.197 3
.320 4
.374
Compute and interpret the mean of the random variable X.
Compute the variance of random variable X.
Compute the standard deviation of random variable x.
What is the probability that a randomly selected student has a parent involved in 3 activities.
What is the probability that a randomly selected student has a parent involved in 3 or 4 activities.

30. A life insurance company sells a $250,000 1-year term life insurance policy to a 20-year old female for $200. According to the National Vital Statistics Report, 56(9), the probability that the female survives the year is compute and interpret the expected value of this policy to the insurance company.